A Quaternionic Proof of the Representation Formula of a Quaternary Quadratic Form

The celebrated Four Squares Theorem of Lagrange states that every positive integer is the sum of four squares of integers. Interest in this Theorem has motivated a number of different demonstrations. While some of these demonstrations prove the existence of representations of an integer as a sum of four squares, others also produce the number of such representations. In one of these demonstrations, Hurwitz was able to use a quaternion order to obtain the formula for the number of representations. Recently the author has been able to use certain quaternion orders to demonstrate the universality of other quaternary quadratic forms besides the sum of four squares. In this paper we develop results analogous to Hurwitz’s above mentioned work by delving into the number theory of one of these quaternion orders, and discover an alternate proof of the representation formula for the corresponding quadratic form.